The direct application of the definition of a limit doesn't really allow us to easily solve any limit related problems. Because of this, we need to come up with some shortcuts to help us solve the limit of a function in a more straightforward manner. We've previously discussed that the limit of a function corresponds to the value of the function as it approaches a certain point. Here, we will try to find that value with the help of some simplified mathematical techniques. Theorems will be presented, and examples will be provided to help you better understand the concept. These are some important theorems your teacher will probably ask you to memorize. Take it slow, read through a few times and then try some of the examples!

THEOREM 1 (A Limit is Unique)

This theorem only signifies the uniqueness of a limit of a function f(x) if it exists. In layman's terms, a specific function has a single specific limit as it approaches a specific point. Sounds confusing? Well, we're basically back in Algebra here (if a = b and a = c then b = c). You have to start somewhere right?

THEOREM 2 (Limit of a Linear Function)

If c and d are constants, then \lim_{x \to a}(cx+d)=ca+d.

Example #1

Solve the following limit:

\lim_{x \to 2}(4x-2)

Solution: Applying the above theorem where c=4 and d=2, we have

\lim_{x \to 2}(4x-2)=4(2)-2=8-2=6

.

THEOREM 3 (Limit of a Constant)

If c is a constant, then \lim_{x \to a}c=c
for any real numbers a & c

Example #2

Find the limit of the following:

\lim_{x \to 9} \sqrt{6}

Solution: By the above theorem, it is clear that when c=\sqrt{6}, then:

\lim_{x \to 9} \sqrt{6} = \sqrt{6}

THEOREM 4 (Limit of an Identity Function)

For any real number c:

\lim_{x \to b}x=b

Example 3

Solve the following limit:

\lim_{x \to \sqrt{29}}x

Solution: By Theorem 4, it is clear that when b=\sqrt{29}, then:

\lim_{x \to \sqrt{29}}x = \sqrt{29}

THEOREM 5 (Limit of the Sum of a Function)

\hbox{If } \lim_{x \to a}f(x)=L \hbox{ and } \lim_{x \to a}g(x)=N

then,

\lim_{x \to a}[g(x)+f(x)]=L+N

Theorem 5 can be simplified to: the limit of a sum is equal to the sum of the respective limits if and only if the limit exists. Consider the following example.